Financial Intermediation at Any Scale
For Quantitative Modelling (3/3)
Cours BachelierCharles-Albert Lehalle
Capital Fund Management, Paris and Imperial College, London
IHP, November 18, 2016 to December 4, 2016

CA Lehalle (Cours Bachelier, 2016)

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Motivation

What about the topic “Intermediation on Financial Markets”

I

Since the 2008-2009 crisis legislators’ and regulators’ viewpoint on financial markets changed,

I

They target to monitor and limit the risk taken by the market participants,

I

In one sentence: they want to ensure most participants plays a role of intermediaries , and nothing more.

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The notion of intermediation and the role of banks, investment banks, dealers, brokers, and now insurancecompanies and funds have evolved and continue to evolve;I

important concepts to understand this are: microstructure and infrastructure; they are linked toliquidity .

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These last 10 years, the field of Market Microstructure emerged. Related literature has grown...

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I am convinced financial mathematics can address quite efficiently core concepts, as partly an academic
and partly a professional, I dedicated the last 12 years to understand these changes from a practical and atheoretical viewpoint.

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These sessions will be the occasion to share how, in my opinion, financial mathematics can answer to new
and important questions raised by recent changes.

CA Lehalle (Cours Bachelier, 2016)

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The Standard Pitch(We Will Go Further Than This)

Following the 2008 crisis, the financial system changed a lot:I

“Clients” (from inside or outside) have no more appetite for sophisticated products.

⇒ The system went from a bespoke market to a mass market.
Bespoke means to sell products that are very different: no economies of scale but high margins.
Mass market means a lot of similar products + optimized logistics.
I

Regulators welcome this change because it can prevent an accumulation of risk in inventories (cf. optimized
logistics).

⇒ The G20 of Pittsburgh (Sept. 2009) put the emphasis on inventory control (it is the root of improved
clearing, segregated risk limits, etc).
⇒ Policy makers took profit of two existing regulations (Reg NMS in the US and MiFID in Europe) to pushtoward electronification of exchanges (i.e. improved traceability and less information asymmetry).
I

Technology went into the game. Think about the kind of recent “innovations” (uber, booking.com, M-pesa,
blockchain, etc): it is about disintermediation .

⇒ How do you desintermediate a system made of intermediates?
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Market Microstructure?

Historically, market microstructure stands for not reducing
I

Sellers = Equity Shares and Bonds issuers

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Buyers = investors.

In practice, today, associated topics are
I

Market impact, Fire sales and Flash Crashes

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Auction / Matching mechanisms (Limit Orderbooks, RFQ, conditional / fuzzy matching, etc)

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Optimal trading / Liquidation

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Market Making and High Frequency Trading

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Investment process while taking all this into account

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My Viewpoint on Market Microstructure
I have been Global Head of Quantitative Research at Crédit Agricole Cheuvreux and CIB during years (including
the crisis). I discuss a lot with regulators; previously inside the working group on Financial Innovation of the
ESMA, now inside the Scientific Committee of the AMF. I am now in a large Hedge Fund.
I

From a Financial
Mathematics
perspective, it isnothing more thanadding a variable toour models: theLiquidity .

I

The interactions
between liquidity andother (usual) variables
is far from trivial.

Disclaimer : I express myown opinion and not the oneof any of these institutions.
CA Lehalle (Cours Bachelier, 2016)

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What Will We Do During 3 Lectures + 1 Seminar?
I will not go in the details of the models (except for few of them), because I target to give you enough information
to include liquidity in the models you know better than me.
Hence, I will
+ 18 Nov:
I

Start by the definition of intermediation

I

Focus on the two main Liquidity variables on financial market: inventories and flows+ 25 Nov:
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Show you what Liquidity looks like when we can observe it+ 2 Dec:
I

Underline why market making (inventory keeping) and optimal trading (flow management) are core for the
new role of market participants.

+ 2 Dec [Seminar]:
I

Explain what practitioners are doing.

It is an on-going work
CA Lehalle (Cours Bachelier, 2016)

6 / 85Advertisement For Optimal Trading

Optimal Trading is About To Close The loop

My own viewpoint on optimal trading:I

We have sophisticated (but tractable) methods to optimize the strategy of one agent (investment bank,trader, asset manager, etc) facing a “background noise” (stochastic control is now really mature),

I

These methods are used by practitioners (already three books on this topic [Lehalle et al., 2013],[Cartea et al., 2015], [Guéant, 2016]),I

Differential games, and more specifically mean field games now propose very promising frameworks toreplace most of the background noise by a mean field of explicitly modelled agents:II

to provide robust results for practitioners [Cardaliaguet and Lehalle, 2016],
to obtain meaningful results for policy recommandations [Lachapelle et al., 2016].

Up to now most results on global modelling used a simplification of a reality. Now decisions are modelled andsystematic, why not inject them into a global model?
It should enable you to produce very accurate models and draw powerful conclusions.
I

Beyond optimal trading, these lectures should help you in introducing liquidity in any model of yours: pleaseask question!

CA Lehalle (Cours Bachelier, 2016)

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Outline of the Sessions

1

The Financial System as a Network of Intermediaries

2

Stylized Facts on Liquidity

3

Optimal Trading

4

Conclusion

5

Closing The Loop

CA Lehalle (Cours Bachelier, 2016)8 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)8 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)8 / 85

Risks Transformation as The Primary Role of The Financial System

To understand the interactions between actors of financial markets, a first step is to understand therole of the financial system .
It takes its role at the root of capitalism:I

say you see a shoes shiner at Deli, IndiaI

you pay $1 to have your shoes shined, and you ask to the guy

I“it seems you have around 30 customers each day, it let you with $30 every day, it is a good job.”I

he answers: “not at all, I earn $1 a day... I do not own the brush, its owner loans its to me $29 a day. Since a
brush costs $12 and I need my daily dollar to eat, I will never own one.”→ let’s discuss about microcredit: loan him $12 during 2 days...
You have $30, you can ask to the guy some percents to cover the risk he will not have enough clients. If you arerisk averse, you can even ask for the brush as collateral... A bank can “structures” the loan for you, it will takecare of all the administrative aspects, it is a simple risk transformation (liquidity on you side, business of theshoes shiner side).

CA Lehalle (Cours Bachelier, 2016)8 / 85

Sometimes Hedging is Far from Optimal

Even on liquid stocks and for vanilla options (close to maturity
in this case), hedging can go wrong.
The 19th of July 2012, a trading algorithms bought and soldshares every 30 minutes without any views on its market
impact [Lehalle et al., 2012].
For one visible mistake like this on liquid underlyings of vanilla
products, how many bad sophisticated hedging processes onless liquid (even OTC) markets...
Anonymous continuous hedging of a remaining position
outside of the bank does not mean all is going well.
Nevertheless we have solutions in recent literature:[Guéant and Pu, 2013], [Li and Almgren, 2014].
But nothing more generic, for instance the whole process ofhedging books in presence of wrong way risk is not studied(as far as I know). One step in this direction is[Schied and Zhang, 2013].
CA Lehalle (Cours Bachelier, 2016)

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A Modern Organization For an Intermediary

My advices to an investment bank:I

Net all your books , maintain two opposite positions is costly and risky,I

If you can’t it may be because you do not communicate enough internally (sometimes because of Chinese
walls...), hence be ready to hedge on the market ,I

But before try to match your small metaorders : send them to an internal place and cross them as muchas possible;

I

You will have synchronization issues (at the level of these metaorders, no reason to be synchronized), ask toyour traders to implement facilitation-like market making schemes inside the bank.

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The remaining quantity has to be sent to markets as smoothly as possible, but it does not mean you will
have no impact. Who is your counterpart in the market should be an obsession: if you trade a one way
risk, you will pay for this in the future...

CA Lehalle (Cours Bachelier, 2016)10 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)11 / 85

The Four Liquidity Variables

Intraday Seasonalities Essentials

CA Lehalle (Cours Bachelier, 2016)

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Volumes are U-shaped, log-volumes are close toGaussian,I

Volatility are U-shaped too (less intense at the end
than at the start of the day),I

Volatility is “more path dependent” than volumes,

I

BA-spread is large at the start of the day, butfinishes small because of market maker running to
get rid of their inventory passively,

I“Volume on the Book” (i.e. Q A + Q B )/2)
seasonality is the invert of the one of BA-spread.
The more the spread is constraied by the tick, themore the seasonality is strong on thevolume-ob-the-book.

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News implies peaks of volume / volatility,

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Activity on other markets has an influence.
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Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)12 / 85

Optimal Trading
Optimal trading is about optimizing a trading process.
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Once a decision to buy or sell has been taken (i.e. a metaorder has been issued),

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Market impact and trading costs can be minimized .

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The process of inserting and cancelling orders in orderbooks has to be automated.

Asset managers delegate to their dealing desk the process of buying or selling. They give to the trading desk
specifications (speed, exposure, etc) to be fulfilled. For instance, some metaorders have to be executed fast,others more slowly.
The dealing desks are reporting to portfolio managers information about liquidity of traded instruments (to takeliquidity into account during the allocation process). And give them “technical” advices, like effects of mergers,
closed days, potentially cheaper instruments.
Specific participants (like high frequency traders of fast hedge funds) take decisions because of
liquidity signals . In their case the buy / sell decision is intricated with their trading process.
Last but not least Market Makers use optimal trading to provide liquidity to other market participants.

CA Lehalle (Cours Bachelier, 2016)12 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)13 / 85

Trading on an Exchange
I

Batch (Fixing) auctions or Continuous auctions,I

price driven or order driven logic,Ibilateral or multilateral trading.

A Limit Order Book (LOB) hosts multilateral, order driven, continuous double auctions.
CA Lehalle (Cours Bachelier, 2016)13 / 85

The emergence of a new market structure

The trading now takes place on a distributed network of heterogenous trading platforms.
CA Lehalle (Cours Bachelier, 2016)14 / 85

What About Multiple Venues?

Fragmentation is the natural counterpart of competition : if you want the exchanges (i.e. trading venues) tocompete.

CA Lehalle (Cours Bachelier, 2016)15 / 85

What About Multiple Venues?

Fragmentation is the natural counterpart of competition : if you want the exchanges (i.e. trading venues) tocompete.

Have you ever seen a mailbox in France?

CA Lehalle (Cours Bachelier, 2016)15 / 85

What About Multiple Venues?

Fragmentation is the natural counterpart of competition : if you want the exchanges (i.e. trading venues) tocompete.

Have you ever seen a mailbox in France?

CA Lehalle (Cours Bachelier, 2016)

This is competition (and fragmentation)

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A typical fragmented stock

The Fragmentation of Microsoft the last 20 days (Source: Fidessa’s fragulator)

CA Lehalle (Cours Bachelier, 2016)16 / 85

A typical fragmented stock

The Fragmentation of Microsoft the last 20 days (Source: Fidessa’s fragulator) more...

CA Lehalle (Cours Bachelier, 2016)16 / 85

Optimal Trading 101: Smart Order Routing (Limit Orders)
Smart Routing of Limit Orders
I

A limit order (i.e. buy –resp. sell– order at a price lower than the best ask –resp. best bid–). It has to be split
too.

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There is structurally uncertainty on limit orders splitting, waiting on a bad queue generates
opportunity costs .

Say on venue n you seen a queue of size Qn and you expect the consuming flow to follow a Poissonprocess with intensity Λn .
P
Goal Queue Jumping: Choose (q1 , ·, qN ) so that the full quantity Q ∗ = n qn is on average executed as fast as
possible.
R tnn
I It means to focus on tn such that
0 dNt = qn + Qn , which expectation reads tn Λn = qn + Qn , whereP∗
n qn = Q .
I

I∗
We do not need a Lagrangian
here,
to note minimizing the minimum of all tn implies
P
Pit is enoughP
P t = tn forany n. Hence t ∗ = Q ∗ / n Λn + n Qn / n Λn . With the convenient notations ρn := Λn /( ` Λ` /N) andPQ̄ := n Qn /N, we obtainqn∗ = ρn
CA Lehalle (Cours Bachelier, 2016)Q∗+ (ρn Q̄ − Qn ) .
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What is Dark Routing?

When more than one trading destination are available (ECNs in the US, Multilateral Trading Facilities -MTF- in
Europe):(i)Ieach of them provides a specific flow φt ,Ikeeping ∆T constant over the trading destinations, each liquidity pool will be able to deliver a quantity Di

Dark Pools are specific trading destinations because:Ithey do not provide pre trade transparency about their limit order books

I

you ask for V and you have min(V , Di ) backIthey allow “pegged” orders: you can specify δS rather than a limit price (“pinging” implies ∆T = 0):
Z τ +∆TDi =t=τ

CA Lehalle (Cours Bachelier, 2016)(i)φt (δS) dt

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Stochastic algorithms

I

The stationary solutions of the ODE: ẋ = h(x) contains the extremal values of F (x) =I

A discretized version of the ODE is (γ is a step):(1)I

0 h(x) dx

xn+1 = xn + γn+1 h(xn )A stochastic version of this being (ξn are i.i.d. realizations of a random variable, h(X ) = E(H(X , ξ1 ))):
(2)

I

Rx

Xn+1 = Xn + γn+1 H(Xn , ξn+1 )the stochastic algorithms theory is a set of results describing the relationship between these 3 formula andthe nature of γ, H, h and ξ [Hirsch and Smith, 2005], [Kushner and Yin, 2003], [Doukhan, 1994]

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Stochastic algorithms theory can be used when you only have a sequential access to a functional you need tominimize:I

to minimize a criteria E(F (X , ξ1 )) of a state variable X

I

if it is possible to compute:∂F(Xn , ξn+1 )∂X
I the results of the stochastic algorithms theory (like the Robbins-Monro theorem [Pagès et al., 1990]) can beused to study the properties of the long term solutions of the recurrence equation:
H(Xn , ξn+1 ) :=Xn+1 = Xn + γn+1 H(Xn , ξn+1 )CA Lehalle (Cours Bachelier, 2016)

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Learning to Capture Flow Optimally in The DarkI

at high frequency, historical statistics are not so usefulI

the limit price S and the quantity V are random variables,

the executed quantity on dark pools has to be maximized (it is market impact free) and sometimes fees aredifferent; this effect is modelled by a discount factor θi ∈ (0, 1) (normalized with respect to a “reference” Litpool)
PN
I the quantity V is split into N parts (one for each DP): r × V is sent to the ith DP (i
i=1 ri = 1)
I

See Optimal split of orders across liquidity pools: a stochastic algorithm approach [Pagès et al., 2011] for moredetails.

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Cost of the executed order

The remaining quantity is to be executed on a reference Lit market, at price S.

The cost C of the whole executed order is given by

C

=

S

N
X

V−

θi min (ri V , Di ) + S

i=1

=

S

V−

N
X

!
min (ri V , Di )

i=1
N
X

!
ρi min (ri V , Di )

i=1

where
ρi = 1 − θi ∈ (0, 1), i = 1, . . . , N.

CA Lehalle (Cours Bachelier, 2016)22 / 85

Mean Execution Cost

Minimizing the mean execution cost, given the price S, amounts to:

Maximization problem to solve

(3)

max

( N
X

)
ρi E (S min (ri V , Di )) , r ∈ PN

i=1

where PN :=nr = (ri )1≤i≤N ∈ RN
+|

PN

o
.

i=1 ri = 1

e := V S and
It is then convenient to include the price S into both random variables V and Di by considering V
e i := Di S instead of V and Di . Assume that the distribution function of D/V is continuous on R+ . LetDP
ϕ(r ) = ρ E (min (rV , D)) be the mean execution function of a single dark pool (Φ = i ϕi (ri )), and assume thatV > 0 P − a.s. and P(D > 0) > 0
Skip Details

CA Lehalle (Cours Bachelier, 2016)23 / 85

The dynamical aspect
We consider the sequence Y n := V n , D1n , . . . , DNn n≥1 .
We will take two types of stationarity assumptions on the sequence(IID) ≡

(ERG)ν ≡



 (i)

the sequence (Y n )n≥1 is i.i.d. with distribution
N+1
ν = L(V , D1 , . . . , DN ) on RN+1
+ , B(R+ ) .

V ∈ L2 (P).

(ii)


(i)









CA Lehalle (Cours Bachelier, 2016)

the sequence (Y n )n≥1 is averaging i.e.
n
(RN+1
1X
+ )
δ(Y k ) =⇒ ν = L(V , D1 , . . . , DN ),P-a.s.
n
k =1

(ii)

supn E(V n )4 < +∞.

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The Lagrangian Approach
We aim at solving the following maximization problem(4)

max Φ(r )r ∈PN

The Lagrangian associated to the sole affine constraint is(5)

L(r , λ) = Φ(r ) − λNX!
ri − 1i=1

So,∀i ∈ IN ,∂L= ϕ0i (ri ) − λ.
∂riThis suggests that any r ∗ ∈ arg maxPN Φ iff ϕ0i (ri∗ ) is constant when i runs over IN or equivalently ifN(6)∀i ∈ IN ,ϕ0i (ri∗ ) =1 X 0 ∗ϕj (rj ).
N
j=1

CA Lehalle (Cours Bachelier, 2016)25 / 85

Existence of maximum

To ensure that the candidate provided by the Lagragian approach is the true one, we need an additional
assumption on ϕ to take into account the behaviour of Φ on the boundary of ∂PN .

Proposition 1
Assume that (V , Di ) satisfies upper assumptions for every i ∈ IN . Assume that the functions ϕi satisfy thefollowing assumption


1
(7)
(C) ≡ min ϕ0i (0) > max ϕ0i
.
i∈IN
i∈IN
N −1
Then arg maxHN Φ = arg maxPN Φ ⊂ int(PN ) where
arg max Φ = r ∈ PN | ϕ0i (ri ) = ϕ01 (r1 ), i = 1, . . . , N .
PN

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Computational Trick

Note thatmaxNX

ρi E(min(ri V , Di )),i=1

X

ri = 1i⇐∀i : E(ρi V 1ri V <Di ) = λ⇔∀i : E(ρi V 1ri V <Di ) =1 X
E(ρj V 1rj V <Dj )N
ji.e. “if each f (i) = λ then each of them equals their average”.

CA Lehalle (Cours Bachelier, 2016)27 / 85

Design of the stochastic algorithm
Using the representation of the derivatives ϕ0i yields that, if Assumption (C) is satisfied, thenCharacterization of the solution

N
X
1
n
o
 = 0.
r ∈ arg max Φ ⇔ ∀i ∈ {1, . . . , N} , E V ρi 1{r ∗ V <D } −ρj 1 ∗
rj V <Dj
i
i
PN
N
 

∗

j=1

CA Lehalle (Cours Bachelier, 2016)28 / 85

Design of the stochastic algorithm
Using the representation of the derivatives ϕ0i yields that, if Assumption (C) is satisfied, thenCharacterization of the solution

N
X
1
n
o
 = 0.
r ∈ arg max Φ ⇔ ∀i ∈ {1, . . . , N} , E V ρi 1{r ∗ V <D } −ρj 1 ∗
rj V <Dj
i
i
PN
N
 

∗

j=1

Consequently, this leads to the following recursive zero search procedure(8)

rin+1 = rin + γn+1 Hi (r n , Y n+1 ), r 0 ∈ PN , i ∈ IN ,where for i ∈ IN , every r ∈ PN , every V > 0 and every D1 , . . . , DN ≥ 0,NX1
ρj 1{r V <D } Hi (r , (V , D1 , . . . , VN )) = V ρi 1{ri V <Di } −
j
j
N


j=1

CA Lehalle (Cours Bachelier, 2016)28 / 85

Back to the Notation of Stochastic Algorithms

When we design a procedure using H(r , ξ1 ), we potentially converge to the extrema of F (r ) :=h(r ) := Eξ1 H(r , ξ1 ).
Here ξt := (V (t), D1 (t), . . . , DN (t)), hence we can use the follwing stochastic procedure:ri (t + 1)==

R

h(r ), whereri (t) + γ(t) · Hi (r (t), (V (t), D1 (t), . . . , VN (t)))


N
1 X

ρj 1{r (t)V (t)<D (t)} ri (t) + γ(t) · V (t) · ρi 1{ri (t)V (t)<Di (t)} −
j
j
N
j=1

⇒ (r1 (∞), · · · , rN (∞)) will be our solution, i.e. the optimal split between Dark Pools .

CA Lehalle (Cours Bachelier, 2016)29 / 85

Back to the Notation of Stochastic Algorithms

When we design a procedure using H(r , ξ1 ), we potentially converge to the extrema of F (r ) :=h(r ) := Eξ1 H(r , ξ1 ).
Here ξt := (V (t), D1 (t), . . . , DN (t)), hence we can use the follwing stochastic procedure:ri (t + 1)==

R

h(r ), whereri (t) + γ(t) · Hi (r (t), (V (t), D1 (t), . . . , VN (t)))


N
1 X

ρj 1{r (t)V (t)<D (t)} ri (t) + γ(t) · V (t) · ρi 1{ri (t)V (t)<Di (t)} −
j
j
N
j=1

⇒ (r1 (∞), · · · , rN (∞)) will be our solution, i.e. the optimal split between Dark Pools .

The underlying idea of the algorithm
is to reward the dark pools which outperform the mean of the N dark pools by increasing the allocated volume
sent at the next step (and conversely).

CA Lehalle (Cours Bachelier, 2016)29 / 85

Constraint Problem

In this algorithm, we took into account the constraintNX

ri = 1,i=1

but notri > 0, ∀1 ≤ i ≤ N.
So the algorithm may exit from the simplex PN stable. To overcome this problem, we have two possibilities
1. Use a Lyapunov function and a strong mean-reverting assumption out of PN : this solution is simpler from amathematical point of view.
2. Force the coefficients ri to stay in PN by a truncation-projection procedure: this solution is more efficient for
applications.

CA Lehalle (Cours Bachelier, 2016)30 / 85

Theorem(s)Theorem 1: Convergence
Assume that (V , D) satisfy upper assumptions, that V ∈ L2 (P) and that Assumption (C) holds. Let
γ := (γn )n≥1 be a step sequence satisfying the usual decreasing step assumptionXγn = +∞

and

n≥1

X

γn2 < +∞.

n≥1

Let (V n , D1n , . . . , DNn )n≥1 be an i.d.d. sequence defined on a probability space (Ω, A, P). Then, there
exists an argmaxP Φ-valued random variable r ∗ such thatNr n −→ r ∗CA Lehalle (Cours Bachelier, 2016)a.s.

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Stochastic Algorithm for Optimal Trading (Conclusion)

We have seenI

how to improve a deterministic simple rule / optimization to a stochastic one.

I

If you write properly the criterion to minimize / maximize, and if you can observe its derivativeIthen it is valuable to build a rigorous stochastic algorithmI

to make the balance between exploration and exploitation.

In the case of Dark trading two other proposals:
I[Ganchev et al., 2010] uses cencored statistics to establish a robust confidence interval [Dimin , Dimax ] aroundeach Di , and then perform a determinstic optimization on (D1min , . . . , DNmin );I[Agarwal et al., 2010] uses minimum regret and maintain a huge table of available quantities (and notproportions).

Moreover you can have a similar approach to choose the price to post in a limit orderbook (cf. learning by trading[Laruelle et al., 2013]).

CA Lehalle (Cours Bachelier, 2016)32 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)33 / 85

Trading Algorithms: Typical Features

BenchmarkType of stock

Type of trade

Main feature

PoV

Medium to large
market depth(1) Long duration position(1) Follows current market flow, (2) Very reactive, canbe very aggressive, (3) More price opportunity driven
if the range between the max percent and min percent
is large

VWAP / TWAP

Any market depth(1) Hedging order, (2) Long duration position, (3) Unwind tracking error (delta hedging of a fast evolving inventory)(1) Follows the “usual” market flow, (2) Good if market
moves with unexpected volumes in the same direction
as the order (up for a buy order), (3) Can be passive

Implementation
Shortfall (IS)Mediumdepthliquidity(1) Alpha extraction, (2) Hedge of a non-linear position(typically Gamma hedging), (3) Inventory-driven trade(1) Will finish very fast if the price is good and enoughliquidity is available, (2) Will “cut losses” if the pricegoes too far away

Liquidity Seeker

Poorafragmented
marketdepth(1) Alpha extraction, (2) Opportunistic position mounting, (3) Already split / scheduled order(1) Relative price oriented (from one liquidity pool toanother, or from one security to another), (2) Capture
liquidity everywhere, (3) Stealth (minimum information
leakage using fragmentation)

CA Lehalle (Cours Bachelier, 2016)33 / 85

Trading Algorithms: Typical Uses

BenchmarkRegion of preference

Order characteristics

Market context

Type of hedged risk

PoV

Asia

Large order size (more than 10%
of ADV: Average daily consolidated
volume)Possible negative news

Do not miss the rapid propagationof an unexpected news event (especially if I have the information)

VWAP / TWAP

Asia and Europe

Medium size (from 5 to 15% of ADV)

Any “unusual” volume is negligible

Do not miss the slow propagation of
information in the market

Implementation
Shortfall (IS)Europe and US

Small size (0 to 6% of ADV)

Possible price opportunities

Do not miss an unexpected pricemove in the stock

Liquidity Seeker

US (Europe)

Any size

The stock is expected to “oscillate”
around its “fair value”

Do not miss a liquidity burst or a relative price move on the stockMore on all this in the three “reference books” for practitioners:
I

Market Microstructure in Practice [Lehalle et al., 2013]I

The Financial Mathematics of Market Liquidity [Guéant, 2016]

I

Algorithmic and High-Frequency Trading [Cartea et al., 2015]

CA Lehalle (Cours Bachelier, 2016)34 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)35 / 85

Optimal Trading Rate

The first papers [Almgren and Chriss, 2000], [Bertsimas and Lo, 1998], focussed on the optimal trading rate, ortrading speed (i.e. how many shares to buy or sell every 5 minutes) for long metaorders.
I

it does not deal with microscopic orderbook dynamics,I

it is a convenient way to take into account any information or constraint at this time scale.

It is very useful for asset managers, brokers, or hedgers. I.e. especially when the decision step is separated fromthe execution step.
Nevertheless it can be used for opportunistic trading too, when risk management at an intraday scale isimportant.

CA Lehalle (Cours Bachelier, 2016)35 / 85

The Almgren-Chriss framework (simplified) .

I have to buy V ∗ share between time 0 and T ; I assume a regular temporal
grid with a time step of δt (n goesPfrom 1 to N = [T /δt]). My volume is splitted in N slices vn such that n vn = V ∗ ([Almgren and Chriss, 2000])
The price follows a Brownian motion:√(9)
Sn+1 = Sn + α δt + σn+1 δt ξn+1
The additive, temporary only, market impact function is ηn (vn ). Then the total cost is:
(10)

W =

N
X

vn · (Sn + ηn (vn ))n=1

Of course, far more sophisticated models have been studied: [Almgren and Lorenz, 2007], [Almgren, 2009],[Bouchard et al., 2011]

CA Lehalle (Cours Bachelier, 2016)36 / 85

Classical Solution .

When the market impact is linear with respect to the participation rate vn /Vn and proportional to the volatilityP(change of variable xn = N
k =n vn ).
W =V ∗ S0| {z }

immediate costin a "free" world+NX

xn σn ξn +

n=1

|

{z

market risk

}

N
X

vn2
Vn
n=1
{z
}
|
ησn

market impact

For a broker algo, we want to minimize a mean-variance criteria:(11)

Jλ = E(W |V1 , . . . , VN , σ1 , . . . , σN ) + λV(W |V1 , . . . , VN , σ1 , . . . , σN )We obtain a recurrence equation in xn :
σn−1 Vnσn−1 Vnλxn+1 = 1 ++ σn Vn xn −xn−1
σn Vn−1η
σn Vn−1

CA Lehalle (Cours Bachelier, 2016)37 / 85

A New Set of Equations to Account For Uncertainty on (V , σ) .

When the Vn and σn are i.i.d. random variables, and when Vn and ξn are correlated:
E(W ) = v ∗ S0 +

N
X
n=1

V(W )

=

N
X

xn2 σn2 +

n=1
N
X

+

n=1

 
σn
(xn − xn+1 )2
ηE
Vn

N
X


η xn (xn − xn+1 )E(σn ) E

n=1

ξn
Vn



 
σn
η V
(xn − xn+1 )4Vn
2

It is easy to add a lot of other effects like: noise on “expected” market impact, auto-correlations, volume-volatility
coupled-dynamic. No more closed-form formula.

CA Lehalle (Cours Bachelier, 2016)38 / 85

This is The Swiss Knife of Trading Curves

It introduced the idea of optimal trading curves → crucial forrisk control.
A lot of effects can be easily added to the AC framework:

CA Lehalle (Cours Bachelier, 2016)Iseasonalities and predictions of V and σ can be plugged,I

arbitrage opportunities can be added, [Lehalle, 2013]39 / 85

This is The Swiss Knife of Trading Curves

It introduced the idea of optimal trading curves → crucial forrisk control.
A lot of effects can be easily added to the AC framework:Iseasonalities and predictions of V and σ can be plugged,I

arbitrage opportunities can be added, [Lehalle, 2013]

I

E(W |V , σ) + λV(W |V , σ) can be replaced by E(W ) + λV(W ),to take uncertainty into account, [Lehalle, 2008],

I

backtest parametric trading curves directly, etc.

Open questions (within this framework)I

the control is the trading rate,Ithe choice of the criterion can be discussed (PoV, VWAP, TWAP, TC, etc.),I

the variance term has a strong influence,I

what can you choose ? (λ).

CA Lehalle (Cours Bachelier, 2016)

39 / 85

More Within the AC Framework

Why introduce a variance term in the cost function?
Iwith λ = 0 and an explicit (exponential) form of orderbook relaxation, Alfonsi and Schied elaborated in thisdirection ([Alfonsi et al., 2009, Gatheral et al., 2012]). A U-shaped trading curve generally stems from suchchoice (due to market impact relaxation and blindness after the last transaction).

⇒ Market impact decays implies a U-shaped trading rate.
I

The choice of a variance term (instead of any p-variation) can be discussed, too[Labadie and Lehalle, 2014]. Especially if you think intraday price dynamics are more mean reverting thandaily ones.

CA Lehalle (Cours Bachelier, 2016)

40 / 85

Practical Use of Automated Trading: Adding Constraints
4

15

Cum. Target Close with PVol constraint

x 10

The Almgren-Chriss criterion is the Implementation Shortfall (i.e.
W (v )), but other “trading styles” are possible, like VWAP (follow theusually traded volume [Cartea and Jaimungal, 2014]), PoV (follow
the real-time market volume), and Target Close.

cum. shares

10
H=0.55 (p=1.8 start=17 switch=102)
H=0.50 (p=2.0 start=34 switch=94)
H=0.45 (p=2.2 start=50 switch=89)
5

0

10

20

30

40

50
60
pillar number

70

80

90

I

The latter targets the closing fixing, trying to avoid a too large
impact.

I

It limits the volume in the fixing auction at q%,I

And does the remaining in an Almgren-Chriss way on
W (v ) − ST100

In practice users put some participation constraints to their trading flow (i.e. v < ρV ). For the Target Close it
raises an interesting problem, especially if you use an estimate of the future volume or if you want to start you
European trading for sure after the opening of US markets. With Mauricio L, we proposed a model and solved it
for fractional Brownian motions [Labadie and Lehalle, 2014].
CA Lehalle (Cours Bachelier, 2016)41 / 85

The Cartea and Jaimungal Version

The usual (simplistic) example of (continuous time) optimal trading
1. Write the Markovian dynamics or the price P, the quantity to trade Q and the cash account X for a sell of Q0
shares before t = T (control is the trading speed r )dQ = −ν dt, dX = r (P − κ · ν)dt, dP = µ dt + σ dW .
2. Write the cost function to minimize
Z T
V (t, p, q, x, ν) = E

XT + QT (PT − A · QT ) + φ!
Qτ2 dτ Ft.

τ =t

3. it gives the HJB and its terminal condition V (Tf , . . .) = x + q(p − Aq); (here µ = 0)
0 = ∂t V +CA Lehalle (Cours Bachelier, 2016)

σ2 2∂ V + φ q 2 + minν {−ν∂Q V dt + ν(p − κ · ν)∂X V } .
2 P

42 / 85

The Quadratic Solution is Nice

4. After the change of variable V (t, p, q, x) = x + q p + v (t, q), you haveon0 = ∂t v + φ q + min −ν∂Q v − κν 2 .
ν

5.
6.

The optimal control is ν ∗ = −∂2
Q v /(2κ), and the PDE 0 = ∂t v + φ q + κ(∂Q v ) /(4κ).
When the value function is quadratic: v (t, q) = h0 (t) + q h1 (t) − q 2 h2 (t)/2, you can separate the PDE inthree:


 0
0

0

=
=
=

h00
h10
−2κh20+−+h12
2h1 h2h22

Cartea and Jaimungal (with misc. co-authors) developed this framework for plenty versions: with a (slightly)
different objective function (VWAP, PoV), with permanent market impact µ → µ + ν, with µt any (adapted)
process, etc.

CA Lehalle (Cours Bachelier, 2016)43 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)44 / 85

Optimal Trading Against Permanent Impact: stylized facts

On our database of 300,000 large orders
Market Impact takes place in different phasesI

the transient impact, concave in time,I

reaches its maximum, the temporary impact, at the end of themetaorder,I

then it decays,I

up to a stationary level; the price moved by a permanent shift.

In [Bacry et al., 2015] we studied all the phases, using intraday and daily analysis (for the first time). We
underlined the importance of some “normalization variables”: the uncertainty on the price formation process , thecapability of the orderbook to resist to volume pressure , and the duration of the metaorder.
Following [Waelbroeck and Gomes, 2013] and simultaneously with [Brokmann et al., 2014], we proposed an
explanation of permanent impact .

CA Lehalle (Cours Bachelier, 2016)44 / 85

Is it Really Possible to Minimize Permanent Impact?

We used an Hawkes-based toy model to show how the concavity of the market impact and the decay can form.
Nevertheless an external parameter C is needed to control the permanent market impact. In such a framework
the intensity of the Hawkes process can be seen as an implicit inventory of the market makers.
I

Part of the price move while an asset manager is buying is due to its trading activity,I

But evidences on the permanent components could be explained by an informational effect:
you buy because you anticipated the price will move. Buy or not: it will move in any case!

This effect seems to have been identified by Waelbroeck and Gomes on “cash trades”, by the CFM team on daily“deconvoluted trades”, and by us on the idiosyncratic component of price moves.
I

The best way to take such permanent market impact into account could simply be to add a deterministic
trend to price dynamics (i.e. “usually, when I decide to buy, the price will go up that way”)... Beside, it could
justify the use of a variance term in the cost function.

This kind of analysis is linked with a potential understanding of the whole market dynamics and the way prices
form.

CA Lehalle (Cours Bachelier, 2016)45 / 85

Optimal Trading Rate (Conclusion)
We have seenI

Almgren and Chriss framework is simple and flexible . It can be seen as a way to include any statisticalproperty of medium-term (5 to 30 min) dynamics in your trading style,I

It does not deal with orderbook dynamics . See [Guéant, 2016] for a lot of variations.

I

It gives a deterministic trading rate .

• But it does not say how to choose the risk aversion . This parameter control the urgency of trading.
I

With small changes to the criterion, Cartea and Jaimungal provide a way to obtain astochastic trading rate . See [Cartea et al., 2015] for variations.
Because the change affects the (not so well defined) risk aversion, it is not that a big deal.

⊕ Risk aversion is a way to deal with the informational component of permanent market impact . See[Lehalle et al., 2013] for a global overview.
⇒ 3 books for optimal trading...

CA Lehalle (Cours Bachelier, 2016)46 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)

47 / 85

Optimal Control of Trading Robots

In practice if you want to rely on previous frameworks (i.e. optimal trading rates), you need a way to realize it
thanks to (optimal?) interactions with orderbook dynamics.
1. Use your favorite method to implement a solver and obtain (in real time) your optimal trading rate ,2. Have statistics ready on a server with the needed parameters (market impact model, risk aversion, calendar
with scheduled events, etc).
⇒ At any point in time, you know the trading rate νt δt you plan to obtain in the next δt seconds.
3. Give it to another logic, focussed on the short term and taking profit of orderbook dynamics. I will call thislogic a Trading Robot , and a priori I will assume it should adopt an exploration-exploitation scheme.
Typically I will use a stochastic algorithm to design my Trading Robot and thanks to backtests and production
results I will have a clear idea of its “usual returns”. I.e. when I ask to Robot A to buy νt δt shares in δt on thiskind of instrument, it obtain Q shares with a price improvement of ∆P (two random variables).

CA Lehalle (Cours Bachelier, 2016)

47 / 85

Optimal Trading Against Orderbook Dynamics

This idea of a decision process in two scales is close to reality:
1. the investor takes a decision according to its views on theprice / risk, but he does not buy or sell himself;
2. he delegates to a executing broker or to a trading algorithm
the trading process.
At the scale of the large order itself there is a similar split in two
scales:CA Lehalle (Cours Bachelier, 2016)Ia scheduler or a human trader takes care of a trading curve(close to the outcome of an Almgren-Chriss optimization),I

it uses trading robots for high frequency interactions withthe orderbook.

48 / 85

Optimizing a Two-Scales Controller

The control
ν = (τiν , δiν , Eiν )i ,In [Bouchard et al., 2011] , we developed a model describing this two
scales process:

The dynamicsR
X ν (t) = x0 + s≤t (b dt + σ dW )P
+ i β(τiν , δiν , Eiν )1τ ν ≤t ,
i

The
Xgain
f (X (τiν + δiν ), Eiν ) + g(X ν (T )).
τiν <TCA Lehalle (Cours Bachelier, 2016)I

The Scheduler launches trading robots, known in probabilistic
terms (the joint laws of their duration and their efficiency) at any time.

I

It is an impulse control problem embedded in a continuous-timeframework: the controls are the stopping times at which robots arelaunched and the quantities given to the robots. The scheduler has
to wait the max between a given duration and the end of the robot
work before launching the next one.

49 / 85

Optimizing a Two-Scales Controller

In [Bouchard et al., 2011] , we developed a model describing this two
scales process:I

The Scheduler launches trading robots, known in probabilistic
terms (the joint laws of their duration and their efficiency) at any time.

I

It is an impulse control problem embedded in a continuous-timeframework: the controls are the stopping times at which robots arelaunched and the quantities given to the robots. The scheduler has
to wait the max between a given duration and the end of the robot
work before launching the next one.

The outcome of this work has been a better understanding of the discretization bias. It is probably not reallypossible to go further with the trading rate as control.
Of course you can try to control directly (and solely) the interactions with orderbooks. We did it in several papers[Guéant et al., 2012, Guéant et al., 2013, Guéant and Lehalle, 2015] . I am not sure it is the best in terms ofrisk control (from an operational perspective at least).

CA Lehalle (Cours Bachelier, 2016)49 / 85

Conclusion on Optimal Trading
We have seen optimal trading has two natural scales:I

At small scales (less than 1 minute on liquid instruments, but could be 1h on not liquid ones) you have totake into account the temporary variations of liquidity.
An exploration-exploitation scheme has to be used. That for you need to
1. Write your criterion,
2. Derive its “mean field” gradient descent / ascent,3. Convert it in its stochastic counterpart (and be sure it converges).

We did it for a Dark SOR.
I

At longest scales (few hours), the strategic interactions between liquidity, urgency, and risk can be
optimized too.
The basic toolkit is the Almgren and Chriss one (it is a mean-variance scheme); it give birth to deterministic
strategies. If you want to do something more subtle, the Cartea and Jaimungal toolkit is nice too, but a little
more complex since it give birth to stochastic optimal strategies.

In practice you need to combine the two scales. The more explicitly, the better. Please do not write only one of
the two framework properly, without paying attention to the other.
At least theoretically, you can try to optimize the all scales simultaneously. It is interesting to understand how theyplay with each others. But in terms of robustness I would recommand to split a trading algorithm in two parts:
one strategic layer (slow and robust) and one tactical layer (fast and adaptive).
CA Lehalle (Cours Bachelier, 2016)

50 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)

51 / 85

Takeaway Message 1: The Financial SystemI

The whole financial system is a partially connected network of intermediaries,

I

They buy and sell risks, netting internally as exposure as possible,Iand using hedging (risk replication) techniques for the remaining, in the hope to cross anotherintermediary’s risk on anonymous markets.

I

The main activity of the financial system is hence to make the market, in an attempt to make margin whileconveying a flow of transactions from some ultimate risk takers (like retail, corporate, asset managers,
insurances) to others.

I

Regulators have to calibrate the amount of risk an intermediary can bear: large enough to enable her towait of next buyers or sellers (to balance her book), small enough to not allow directional risk.

I

More continuous the trading, easier to balance the book and achieve very low levels of risk for the wholesystem.

CA Lehalle (Cours Bachelier, 2016)

51 / 85

Takeaway Message 2: Optimal Trading

I

Optimal trading covers: execution of metaorders, opportunistic liquidity trading and market making.

I

It makes the balance between market impact (trade slow) and the inventory risk (trade fast).

I

At the smallest scale, I recommend the use of forward methods, like statistical learning. I have shown
how stochastic algorithms could offer more accurate updates of variables of interest than reinforcement
learning.

I

At the largest scales, stochastic control is needed since the backward component is crucial (do not missthe target).

I

You can try to optimize the two layers at once, as a practitioner I would not recommend this for robustness
reasons (large exposure to small perturbations of HF data). Here the design of interactions between theslow layer (implementing backward strategies) and the fast one (focussed on tactical forward methods) in aninteresting algorithmic challenge.

CA Lehalle (Cours Bachelier, 2016)52 / 85

Takeaway Message 3: Stakes for financial maths

I

introduce liquidity in existing models

I

it means: instantaneous, transient, decay and permanent market impact

Ior offer and demand (multiple agents)I

introduce fragmentationI

it means a choice between multiple "liquidity shapes"

I

introduce learning

I

dynamics of liquidity is sophisticated (cf the Queue Reactive model)I

it predicts short term price movement

Competition for liquidity is clearly an important topic.

CA Lehalle (Cours Bachelier, 2016)

53 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)

54 / 85

Different Levels of Loops to Close

Two areas are not explored enoughIfor practitioners : statistical learning; how to adapt online to regime switches (remember what we said
about liquidity game vs. price game)? How to be robust to transitory phases? “Closing the loop” withlearning is mixing exploration and exploitation.

I

for regulators : game theory; what is the result of putting rational agents together? The more quants will
read the 3 books, the more it will be needed to understand such interactions, and how changing “metaparameters” (ie rules) will modify the outcome of this game?

For game theory on financial market:I

few agents usually leads to principal - agent problems,I

a lot of agents usually leads to mean field games.

Moreover, game theory is a way to obtain robust control .

CA Lehalle (Cours Bachelier, 2016)

54 / 85

Mean Field GamesI

the number of players needs to be "large enough"I

all players contribute to a "mean field" (i.e. a global variable: available shares, volatility, resource, etc)Ia function of this mean field (at least its mean, may be its standard deviation, etc) appear in this utility
function of the players→ the name on the player cannot be used, but they can have a parameter (like a time horizon or risk aversion)of their own
The methodology is similar to the one to solve static Nash games:I

express the solution (for one agent) and find the solution as if the mean field was knownI

you obtain a backward pde

Icombine what you know about the mean field to find its forward pde

Liquidity is typically a mean field: the state of the inventory of participants influence their costs and can lead tofire sales [rené]. What practitioners call "velocity" of the liquidity (the flows) is a mean field too, it probably forms
the prices along with market impact.

CA Lehalle (Cours Bachelier, 2016)

55 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)56 / 85

MFG of Controls and An Application To Trade Crowding

Joint work with Pierre
Cardaliaguet

A continuum of agents trade optimally “à la Cartea-Jaimungal”. Main variables:
R a
I public price P, exposed to permanent market impact. µ =q ν df (a) is the sum of the control of all agentsthe remaining qty of each agent Q a (ν a is the control)
the wealth of each agent (public price is penalized by instantaneous impact)
Keep in mind ν a is negative for a seller
I

I

(12)

dSt = αµt dt + σ dWt .

(13)

dQta = νta dt,

since for a seller, Q0a > 0 (the associated control ν a will be mostly negative) and the wealth suffers from linear
trading costs (or temporary, or immediate market impact):

dXta = −νta (St + κ · νta ) dt.

(14)

The cost function of investor a selling from t = 0 and T is similar to the ones used in [Cartea et al., 2015]: theterminal inventory is penalized and a quadratic running cost is subtracted:
!
Z T
a
a
a
a
a
a
a 2
(15)
Vt := sup E XT + QT (ST − A · QT ) − φ(Qs ) ds Ft .
νCA Lehalle (Cours Bachelier, 2016)s=t

56 / 85HJB For One Player

The Hamilton-Jacobi-Bellman associated to (15) is1
0 = ∂t V a − φa q 2 + σ 2 ∂S2 V a + αµ∂S V a + sup ν∂Q V a − ν(s + κ ν)∂X V a , V a (T , x, s, q; µ) = x + q(s − Aa q).
2
ν
Following the Cartea and Jaimungal’s approach, we will use the following ersatz:
V a = x + qs + v a (t, q; µ).

(16)
Thus the HJB on v isno−αµ q = ∂t v a − φa q 2 + sup ν∂Q v a − κ ν 2 , v a (T , q; µ) = −Aa q 2 .
νand the associated optimal feedback is(17)

CA Lehalle (Cours Bachelier, 2016)ν a (t, q) =∂Q v a (t, q).
2κ

57 / 85

Obtaining The MFG Master Equation
The mean field of this framework is the distribution m(t, dq, da) of the inventories Q0a and of their preferences(φa , Aa ).
It is then straightforward to write the net trading flow µ at any time t
Z
Z∂Q v a (t, q)
(18)
µt =
m(t, dq, da).
νta (q) m(t, dq, da) =
2κ
(q,a)
q,a
v a is an implicit function of µ, meaning we will have a fixed point problem to solve in µ. By the dynamics (13) of
Qta , we have∂ va
= 0 with initial condition m0 = m0 (dq, da).
∂t m + ∂ q m Q
2κ

(19)


Z

 −αq
(q,a)




∂Q v a (t, q)m(t, dq, da)
2κ


∂ va

Q
∂t m + ∂q m 2κ=∂ t v a − φa q 2 +=0Terminal conditions: m(0, dq, da) = m0 (dq, da),CA Lehalle (Cours Bachelier, 2016)(∂Q v a )24κv a (T , q; µ) = −Aa q 2 .
58 / 85

The System in The Case of Identical Preferences and Quad. v φa ≡ φ, Aa ' A
Z
Set E(t) = E [Qt ] =qm(t, dq). Note thatq

(20)

E 0 (t) =Zq∂t m(t, dq), E 0 (t) = −
q

Z
q∂q
q



Z
∂Q v (t, q)∂ v (t, q)dq =m(t, dq).
m(t, q) Q2κ2κqWhen v (t, q) can be expressed as a quadratic function of q: v (t, q) = h0 (t) + q h1 (t) − q 2Z(21)µ(t) =qand
E 0 (t) =∂Q v (t, q)
h1 (t)h2 (t)
dm(q) =
−
E(t).
2κ
2κ
2κ


Z
m(t, q)qh2 (t),2h2 (t)
h1 (t)
−
q
2κ
2κ


dq =

h1 (t)h2 (t)
−
E(t).
2κ
2κ

So we can supplement it with
(22)

CA Lehalle (Cours Bachelier, 2016)2κE 0 (t) = h1 (t) − E(t) · h2 (t).

59 / 85

Master Equation For Identical Pref. and Quad. v

We now collect all the equations. Recalling (21), we find:4κφ = −2κh20 (t) + (h2 (t))2 ,
(23a)





αh2 (t)E(t) = 2κh10 (t) + h1 (t) (α − h2 (t)) ,
(23b)


− (h1 (t))2 = 4κh00 (t),
(23c)




2κE 0 (t) = h1 (t) − h2 (t)E(t).
(23d)
with the boundary conditions
h0 (T ) = h1 (T ) = 0, h2 (T ) = 2A, E(0) = E0 ,where E0 =density m).

R

q qm0 (q)dq is the net initial inventory of market participants (i.e. the expectation of the initial

CA Lehalle (Cours Bachelier, 2016)60 / 85

Solving The Mean Field
To summarize, the equation satisfied by E is:
0 = 2κE 00 (t) + αE 0 (t) − 2φE(t)for t ∈ (0, T ),(24)
E(0) = E0 ,
κE 0 (T ) + AE(T ) = 0.
Closed form for the net inventory dynamics E(t)For any α ∈ R, the problem (24) has a unique solution E, given byE(t) = E0 a (exp{r+ t} − exp{r− t}) + E0 exp{r− t}where a is given bya=(α/4 + κθ − A) exp{−θT }
,
−α
sh{θT
} + 2κθch{θT } + 2Ash{θT }2the denominator being positive and the constants rα± and θ being given bysα1α2r± := −
± θ,
θ :=
κφ +
.
4κ
κ
16
CA Lehalle (Cours Bachelier, 2016)

61 / 85

The Standard Case
E(t), E0 = 10.00, α = 0.400

10

T = 5.00, A = 2.50, κ = 0.20, φ = 0.10

5

h2(t)
4

8

−h1(t)

3
6
2
4
1
2

0
0

0

1

2

3

4

5

−1
0

1

2

3

4

5

Dynamics of E (left) and −h1 and h2 (right) for a standard set of parameters: α = 0.4, κ = 0.2, φ = 0.1,
A = 2.5, T = 5, E0 = 10.
CA Lehalle (Cours Bachelier, 2016)62 / 85

Small α105h2(t) ref. case

E(t) ref. case
E(t) small α4−h1(t) ref. caseh2(t) small α836−h1(t) small α

2

1

4

0
2
−1
0
0

1

2

3

4

5

−2
0

1

2

3

4

5

Comparison of the dynamics of E (left) and −h1 and h2 (right) between the “reference” parameters of Figure ??
and smaller α (i.e. α = 0.1 instead of 0.4) such that |h1 (0)| is smaller.
CA Lehalle (Cours Bachelier, 2016)

63 / 85

h2 Almost Constant
105h2(t) ref. case

E(t) ref. case√
E(t) A ' κφ8

4−h1(t) ref. case√h2(t) A ' κφ√−h1(t) A ' κφ

3
6
2
4
1
2

0
0

0

1

2

3

4

5

−1
0

1

2

3

4

5

Comparison of the √dynamics of E (left) and −h1 and h2 (right) between the “reference” parameters of Figure ??
and when κφ ' A: in such a case h2 is almost constant but E and h1 are almost unchanged.
CA Lehalle (Cours Bachelier, 2016)64 / 85

A Case With Not Monotonous EE(t), E0 = 10.00, α = 0.010

10.0

30

T = 5.00, A = 2.50, κ = 1.50, φ = 0.03
h2(t)

20
9.5

−h1(t)

10
0
−10

9.0

−20
−30

8.5

−40
8.0
0

1

2

3

4

5

−50
0

1

2

3

4

5

A specific case for which E is not monotonous: α = 0.01, κ = 1.5, φ = 0.03, A = 2.5, T = 5 and E0 = 10.
CA Lehalle (Cours Bachelier, 2016)65 / 85

Not Monotonous E
φ ' 0.003

1.0
0.8

κ

0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

φ ' 0.026

1.0
0.8

κ

0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

α

0.8

1.0

5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

φ ' 0.011

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

φ ' 0.034

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

α

0.8

1.0

5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

φ ' 0.018

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

φ ' 0.042

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

α

Numerical explorations of t m for different values of φ (very small φ at the top left to small φ at the bottom right) on
the α × κ plane, when T = 5 and A = 2.5. The color circles codes the value of t m : small values (dark color)when E changes its slope very early; large values (in light colors) when E changes its slope close to T .
CA Lehalle (Cours Bachelier, 2016)

66 / 85

Conclusion on MFG of Controls For Liquidation

It is a proof of maturity of the use if stochastic control in financial math:I

Four years ago, it was difficult to think about a game theoretical version of the Almgren and Chriss optimalliquidation problem (schied and jaimungal).

I

Our understanding of the problem itself improved (see Guéant and Cartead and Jaimungal books)

Iand some extensions of MFG have been needed (see the paper).

I

but we now know how to handle it (and in a specific case it is fully solved)

Solving game theoretical versions of what we know is important (instead of sophisticating it in a mean field becausegame),
I

it is a way to obtain robust control

I

it helps regulator to understand the system to adjust some meta parameters (κ is this example)

MFG is not the only way to answer to such questions.
Moreover learning should not be forgot (done in our paper): what does change when information is not complete?

CA Lehalle (Cours Bachelier, 2016)

67 / 85

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System

2

Stylized Facts on Liquidity

3

Optimal Trading
Learning by Trading (in The Dark)
Trading Benchmarks
Optimal Trading Rate
Optimal Trading Against Permanent Impact: stylized facts
Optimal Control of Trading Robots

4

Conclusion

5

Closing The Loop
MFG of Controls
Kyle’s Model

CA Lehalle (Cours Bachelier, 2016)

68 / 85

Making the Market in Front of Adverse Selection: Kyle’s 85 Model

The framework

CA Lehalle (Cours Bachelier, 2016)

I

An informed trader, knowing the future priceI

Noise traders, knowing nothingI

A market makers, having only access to distributions (thanks to“backtests” / observations); she changes her price linearly according to
the price pressure she observes: fP (q) = P̃ + λ · q.

68 / 85

Making the Market in Front of Adverse Selection: Kyle’s 85 Model

The frameworkI

An informed trader, knowing the future priceI

Noise traders, knowing nothingI

A market makers, having only access to distributions (thanks to“backtests” / observations); she changes her price linearly according to
the price pressure she observes: fP (q) = P̃ + λ · q.

I

The informed trader adjusts his participation to maximize its profit (given P̃
and λ),The informed trader chooses his
quantity to maximize his expected
profitCA Lehalle (Cours Bachelier, 2016)

68 / 85

Making the Market in Front of Adverse Selection: Kyle’s 85 Model

The framework

The market maker choose P̃ andλ to adjust her price to the flow

CA Lehalle (Cours Bachelier, 2016)

I

An informed trader, knowing the future priceI

Noise traders, knowing nothingI

A market makers, having only access to distributions (thanks to“backtests” / observations); she changes her price linearly according to
the price pressure she observes: fP (q) = P̃ + λ · q.

I

The informed trader adjusts his participation to maximize its profit (given P̃
and λ),I

The market makers know the distribution of the informed price and set P̃
and λ so that her price is as close as possible to its expectation.

68 / 85

Making the Market in Front of Adverse Selection: Kyle’s 85 Model

Following Continuous Auctions and Insider Trading – [Kyle, 1985]:
I

Remember the market makers fear adverse selection.

I

We have informed traders, they know the price will be pω after their trade, pω ∼ N (P ∗ , σp2 ).

I

Other traders, (i.e. noise traders for Kyle) trade for other reasons, their net direction is nω ∼ N (0, σn2 ).

I

The informed traders have to choose a participation Q(p) (they know p) to maximize their profit,

I

Knowing the market makers (MM) will react to the net perceived flow linearly: the public price will befP (Q(pω ) + nω ) = P̃ + λ · (Q(pω ) + nω ).

I

Moreover in their filtration, the MM should produce a price being the best estimator of pω given
Q(pω ) + nω .

CA Lehalle (Cours Bachelier, 2016)

69 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q

CA Lehalle (Cours Bachelier, 2016)70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

CA Lehalle (Cours Bachelier, 2016)70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒
I

They decide to trade Q(pω ) = (pω − P̃)/(2λ).
The MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

CA Lehalle (Cours Bachelier, 2016)70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :CA Lehalle (Cours Bachelier, 2016)70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :CA Lehalle (Cours Bachelier, 2016)


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :CA Lehalle (Cours Bachelier, 2016)


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )

⇒



 P̃

=


 λ

=

P∗
σp2 /(2λ)σp2 /(2λ)2 + σn2

70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :

⇒


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )

⇒



 P̃

=


 λ

=

P∗
σp2 /(2λ)σp2 /(2λ)2 + σn2

It can be solved with λ = σp /(2σn ).

CA Lehalle (Cours Bachelier, 2016)70 / 85

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :

⇒


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )

⇒



 P̃

=


 λ

=

P∗
σp2 /(2λ)σp2 /(2λ)2 + σn2

It can be solved with λ = σp /(2σn ).

I

The more potential informational price move (i.e. large σp ), the largest impact.

I

The more non informative flow, the more difficult for the MM to identify information, hence the less she
impacts the price.

CA Lehalle (Cours Bachelier, 2016)70 / 85

It is a Little More Complex in PracticeOn our database of 300,000 large orders

CA Lehalle (Cours Bachelier, 2016)

I

This is renormalized market impact as a function of the participation rate, i.e.
v /V , taken from [Bacry et al., 2015].

I

It is not linear, it is concave .

71 / 85

An Extended Version of Kyle’s Model
Due to [Çetin and Danilova, 2015].
Notations and definitions.
I

At t = 0 , the insider trader knows the final “fundamental” value of the instrument. This value is drawn from arandom variable V (the law of V is known by everyone). He tries to trade “optimally” knowing the market
maker with react to the trading flow. His cumulated trading flow at t is Xt .

I

Liquidity traders generate a (signed) cumulative demand of σt Bt at t.

I

The Market Maker (in reality N market makers competing “à la Bertrand”) observes the net demand
Yt := Xt + σBt . She sets the price at St := H(t, Yt ). She has a CARA utility function U(x) = − exp −ρx.

Note the terminal wealth of the insider is
Z 1
Z 1
(25)
W1X =
Xt dH(t, Yt ) + X1 (V − H(1, Y1 )) =(V − H)dX .
00

On her side, the market maker wealth at t is
Z 1
1
Y1
(26)
Gt := −
Ys dH(s, Ys ) + 1t=1
(H(1, Y1 ) − V ).
N 0
NCA Lehalle (Cours Bachelier, 2016)

72 / 85

The Market Maker’s Viewpoint

Say the infinitesimal increase of the market maker pricing rule H can be written using a controlled diffusion:dS = Zt dB Y + µt dt, then the infinitesimal change in U is (using (26)):
ρdU(Gt ) = U(Gt ) Nρ Yt σt dB Y + (µt + 2N
Yt Zt2 )dt . The dynamic programming principle states its deterministic
part should be zero, hence(27)µt = −

ρ
YZ 2 .
2N

And finally(28)

dS = Zt dB Y −ρYt Zt2 dt.
2N

Dynamics of the Pricing Rule
Because of the term YdS in the wealth of the market maker and because of her CARA(ρ) utility, the pricing ruleρ
necessarily follows equation(28): dS = Zt dB Y − 2N
Yt Zt2 dt.

CA Lehalle (Cours Bachelier, 2016)

73 / 85

MM Necessary Condition: Backward Heat Guessing

On his side the insider will act on his trading flow dX := α(t, Y , S, Z )dt, the net demand will hence be a diffusion
with a controlled drift:(29)

dY = σt dB + αdt.

Using α, the insider will hence maximize his utility; again the dynamic programming principle onRΨ(t, y ) := supα E t1 (V − H)αds Yt = y gives a useful information: the following HJB (subscripts stand for22partial derivatives): Ψt + σ2 Ψyy + supα {α(Ψy + (V − H))} = 0. In other terms: Ψy = V − H; Ψt + σ2 Ψyy = 0.
This implies H must satisfy the heat equation too:(30)

CA Lehalle (Cours Bachelier, 2016)

Ht +

σ2
Hyy = 0.
2

74 / 85

The Insider’s Viewpoint: Price Dynamics Follows MM’s Willσ2
On the one hand, writing Ito’s formula on S = H(t, Y ) reads dS = Hy dY + (Ht +Hyy )dt = Hy σt dB + Hy αdt.
{z2
}
|
=0

And on the other hand we have (28) because of the market maker optimal strategy. Equalling the deterministic
parts reads
(31)

Hy =

Z
.
σ

and the random parts reads Hy α = −ρYZ 2 /(2N). Hence (z/σ)α∗ (t, y , s, z) = −ρ/(2N) yz 2 , i.e.
(32)

CA Lehalle (Cours Bachelier, 2016)α∗ = −

ρσyz.
2N

75 / 85

The Insider’s Viewpoint: Summary

Insider’s Optimal Strategy
Because of Ito on the pricing rule seen as a function of (t, Y ) and because of the shape of the pricing rule fixedby the market maker, we necessarily have equations (31) Hy = Z /σ and (32) α∗ = −ρσ · yz/(2N). It implies2

equation (33) for the net demand dynamics: dY = σt dB − ρσ
YHy dt.
2N

Terminal Condition
Because of the shape of the insider’s wealthnecessarily have H(1, Y1 ) = V .

R10 (V − H)dX and because the expectation of dY is dX , weRemark: Obtaining the terminal condition is tricky.

CA Lehalle (Cours Bachelier, 2016)76 / 85

Confronting Viewpoints

Replacing α and z by their values in (29) allows to write
dY

=
=

(33)

=

σt dB + αdt
ρσσt dB −
YZ dt, using (32)2N
ρσ 2
YHy dt, using (31).
σt dB −
2N

Putting the main equations (30)-(33) side to side, we now haveσ2
 Ht +
Hyy = 0, under the terminal condition H(1, Y1 ) = V
2
(34)
ρσ 2


 dY = σt dB −YHy dt2N

CA Lehalle (Cours Bachelier, 2016)

77 / 85

How to Solve The Closed Loop
The iterations to solve the closed loop are as follow (use Schauder’s fixed point theorem, i.e. a strict
compactness inclusion argument)
I Initialization.
V is the final value distribution, it is revealed to the insider trader at t = 0.
Typically V ∼ N (0, σV2 ).
I The market maker makes a guess for Y (1) , the sum of noise traders’ flows and the insider’s one.
1
It can be a Gaussian too, or something more “protective” (in the sense a distribution with fat tails).
I She solves (a backward way) the heat equation

I

(1)

Ht

+

σ 2 (1)
H =0
2 yy

(1)

with the terminal condition H (1) (1, Y1 ) = V . She obtain a “first guess” pricing rule to apply.
I

Steps to Iterate.
I

The insider trader now sees H (n) (t, Y (n) ) in real-time, and reacts by setting his flow so that
dY

I

(n+1)

= σn dβ −ρσ 2 (n+1) (n)YHy dt.
2N t

At the end of the day the market maker discovers Y (n+1) and know V in distribution, hence she sets a new pricing rulesolving again the heat equation, but with a different terminal condition, nowHCA Lehalle (Cours Bachelier, 2016)

(n+1)

(1, Y

(n+1)

) = V.
78 / 85

A Qualitative Understanding

This master system of equations can be understood as

I implemented the numerics, here is
the “market impact” (i.e. pricing rule)function (and its evolution during the
tracking of the fixed point). It isconcave, accordingly with the data.

CA Lehalle (Cours Bachelier, 2016)I

The market maker(s) tunes her pricing rule H such that it follows the heat
equation and it exactly fits the law of the fundamental prices V at terminal
time t = 1;

I

The insider mean reverts the net demand (his demand plus the noisy one)flow (i.e. net demand) through time using a mean reversion strength
proportional to ∂y H(t, y )|(t,Yt ) : when the slope of the pricing rule isintense, it mean reverts more than when it is not.

The complexity of this system is the market maker needs to know in advancethe insider’s strategy to have an access to Y1 to satisfy her terminal condition.
That for a “fixed point” theorem will be needed, the good one is Schauder’sone, and that for it will be needed Y has a smooth transition density. This
property will need to be stable via the application of the system (34).

79 / 85

Thank You For Your Attention

CA Lehalle (Cours Bachelier, 2016)

80 / 85References I
Agarwal, A., Bartlett, P. L., and Dama, M. (2010).
Optimal Allocation Strategies for the Dark Pool Problem.
In Teh, Y. W. and Titterington, M., editors, Proceedings of The Thirteenth International Conference on Artificial Intelligence and Statistics (AISTATS), volume 9, pages 9–16.
Alfonsi, A., Fruth, A., and Schied, A. (2009).
Optimal execution strategies in limit order books with general shape functions.
Quantitative Finance, 10(2):143–157.
Almgren, R. (2009).
Optimal Trading in a Dynamic Market.
Technical Report 2.
Almgren, R. and Lorenz, J. (2007).
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